Dynamical Analysis And Control For Several Classes Of Fractional Systems

Fractional calculus is the theory of the differentiation and integration on any order. As an extension of integer-order calculus on arbitrary order, it has aroused considerable attention of scholars both at home and abroad. Fractional calculus has powerful advantages and wide application in many areas, such as physics, neural network, medicine, control engineering and so on. It has been confirmed that time delay is ineluctable in the real networks in terms of various studies. Time delay has an important influence on the dynamic behaviors for the fractional-order systems. Currently, the dynamics of the delayed fractional-order systems has become a hot research topic.On the basis of previous research, this dissertation further discusses the dynamic problems for the fractional-order neural networks, fractional-order gene regulatory networks, fractional-order ecosystem. Presently, the theoretical results on the bifurcation of such fractional-order networks have rarely been obtained. This dissertation focuses on the problem of the stability and bifurcation control for the fractional-order systems with time delays, the main achieve-ments of this dissertation are as follows:1. The issue of stability and bifurcation for the fractional-order neural network with three neurons is discussed. Firstly, a fractional-order neural network model involving three neurons without delay is proposed. Then, the dynamic behaviors of the proposed system are analyzed by means of the matrix eigenvalue theory, bifurcation theory and the stability theory of fractional-order systems. The stability criterion and the conditions of bifurcation are addressed by choosing the order or the system parameter as the bifurcation parameter, respectively. It is found that the order or the system parameters play an important role on the stability and bifurcation of such system. The proposed system is stable when the order is less than the critical value, and the system loses its stability and a Hopf bifurcation occurs when the order passes through the critical value. If the order and other parameters are established, when the system undergoes a Hopf bifurcation, the bigger the system parameter is, the more stable the system is.2. The dynamical behaviors of the delayed fractional-order neural networks are analyzed. The delay-dependent stability criterion is obtained by analyzing the characteristic equation of the linearized system where time delay is chosen as the bifurcation parameter, and the conditions of the occurrence of Hopf bifurcation are given. It is demonstrated that both the order and time delay have an important influence on the dynamic properties for the delayed fractional-order neural network. It is indicated that the order can effect the position of the bifurcation points. The order can advance or delay the onset of bifurcation for different order is selected. The system may maintain stable for the small delay, and result in instability and bifurcation occurs for the large delay.3. The problem of bifurcation for a class of delayed fractional-order ring-structured neural network with arbitrary neurons is investigated. Since the complex neural networks include the order, multiple time delays and many neurons, therefore, the discussed models are much closer to the complex, real network. Firstly, the characteristic equation of the linearized system is studied where the sum of time delays is selected as the bifurcation parameter, and the condition of the Hopf bifurcation is obtained. It is shown that although the delay and the number of neurons are more for the proposed network, its stability and bifurcation of the system is only established by the sum of time delays. Secondly, we find that the order will affect the bifurcation point of the system. The larger the order is, the earlier Hopf bifurcation emerges. If the system parameters are given, the mechanism of influence on the dynamics of the number of neurons for such system is further discussed. It reveals that a Hopf bifurcation will occur when the number of neurons is odd. The more the number of neurons is, the more earlier Hopf bifurcations emerges. The system always maintain unstable and Hopf bifurcation never occurs when the number of neurons is even.4. The dynamical behaviors of a class of delayed fractional-order complex-valued neural network with two neurons are studied. The impact on the stability and bifurcation of time delay for the proposed system is investigated. Based on the assumptions, the complex value neural network is transformed into the real valued one to discuss the dynamical properties for the system. The stability and bifurcation of the system are studied by using time delay as the bifurcation parameter. It reveals that time delay can influence the stability of system. The system is stable when the delay is small. The system will lose stability and Hopf bifurcation occurs when delay exceeds a certain critical value. The conditions of bifurcations are presented. Moreover, it also manifests that the order will affect the location of the bifurcation point. The onset of bifurcation of the system will be postponed as the order gradually increases.5. The issue on bifurcation and control for a class of delayed fractional-order gene regula-tory networks is examined. Firstly, by using the sum of the delay as the bifurcation parameter, the dynamic behaviors of delayed genetic regulatory network are analyzed without control and delay-dependent stability criterion and the condition of bifurcation is derived. It is illustrated that time delay may influence the stability of the network, and the order can effect the location of bifurcation. Secondly, a hybrid controller is designed to control the dynamic properties of such network. The complex dynamic behaviors of the controlled network are investigated. It is depicted that the larger the order is, the smaller the bifurcation point becomes if the feedback gain is fixed, and bifurcation occurs in advance. If the order is given, the smaller the feedback gain is, the larger the stability region is for controlled fractional-order network, the onset of bifurcation is delayed.6. The dynamic properties of the fractional-order delayed predator-prey ecosystem with incommensurate orders are discussed. Firstly, a class of fractional-order delayed predator-prey ecosystem models with different orders is proposed. Then, based on the characteristic equation, time delay is served as the bifurcation parameter, and the effects of time delay on the dynamics for the fractional order ecosystem are analyzed. It is shown that time delay has a great influence on the stability of the proposed system. In addition, the effects of each order on the stability and bifurcation of the system are investigated. Under the conditions of the two fixed orders, the speed of the bifurcation for the proposed system is discussed according to the third order. The results display that the third order has important influence on the speed of bifurcation for such system. The higher the order is, the more bifurcation occurs in advance. Finally, by designing the linear time-delay feedback controller, the onset of bifurcation can effectively be controlled. It is indicated that the smaller the feedback gain is, the better the effect of bifurcation control is.The studies of the present dissertation indicate that the order, the system parameters, time delay and the feedback gain have important influence on the dynamic behaviors of the fractional-order systems. The order can effect the stability of fractional-order systems, can also advance or delay the occurrence of bifurcation for fractional-order systems. One can improve the stability of fractional-order systems by adjusting the system parameters. The impact of time delay of fractional-order system is dual. Time delay can make the fractional-order system lose its stability, and lead to the onset of bifurcation. Under certain conditions, an appropriate time delay can enhance the stability of fractional-order systems. The feedback gain can control the emergence of bifurcation for fractional-order systems. The derived results of this dissertation can not only enrich the theory of nonlinear science, but also provide a theoretical guidance of the application for fractional-order nonlinear system in control engineering.